1 – There is no reason why a pregnant woman should not perform Hajj. A pregnant woman is taahir (pure), she has to pray and fast, and if she is divorced by talaaq, her divorce is acceptable according to the Sunnah.
2 – indeed it is narrated in the Sunnah that Asma’ bint ‘Umays (may Allaah be pleased with her) went out for Hajj with the Prophet (peace and blessings of Allaah be upon him) when she was pregnant and approaching full term, and she gave birth at the meeqaat.
It was narrated that ‘Aaishah (may Allaah be pleased with her) said: “Asma’ bint Umays – the wife of Abu Bakr – gave birth to Muhammad ibn Abi Bakr in Shajarah and the Messenger of Allaah (peace and blessings of Allaah be upon him) told Abu Bakr to tell her to do ghusl and enter ihraam.
Narrated by Muslim, 1209
Shajarah refers to Dhu’l-Hulayfah, i.e., the meeqaat from which the people of Madeenah enter ihraam.
Al-Nawawi said, concerning what we learn from this hadeeth:
This indicates that the ihraam of women who are bleeding following childbirth or who are menstruating is valid and that it is mustahabb for them to do ghusl before entering ihraam. There is scholarly consensus that it is obligatory, but our view and that of Maalik, Abu Haneefah and the majority of scholars is that it is mustahabb. Al-Hasan and the Zaahiri madhhab said that it is obligatory.
All the actions of Hajj on the part of women who are bleeding following childbirth or who are menstruating are valid, apart from Tawaaf and the two rak’ahs following tawaaf, because the Prophet (peace and blessings of Allaah be upon him) said: “Do what the pilgrims do but do not do tawaaf.”
Sharh Muslim, 8/133
If a woman has not yet done the obligatory Hajj, then pregnancy is not an excuse for her not to do Hajj, because she can avoid the places where there is too much crowding and pushing and shoving. If she is unable to stone the Jamaraat herself, she can delegate someone to do so on her behalf. If she cannot do tawaaf and sa’ee, then she can do so in a wheelchair, and so on.
Many people manage to do Hajj in comfort without any hardship.
3 – Yes, if a woman is pregnant and trustworthy doctors tell her that going for Hajj will pose a risk to her or her baby because she is sick or weak or for some other reason, then she should be prevented from doing Hajj that year. This is indicated by the words of the Prophet (peace and blessings of Allaah be upon him), “There should be no harming nor reciprocating harm.” (Narrated by Ibn Maajah, 2340; this is a hasan hadeeth. For details of its isnaad see Jaami’ al-‘Uloom wa’l-Hikam by Ibn Rajab, 1/302)
4 – Some doctors differentiate between early pregnancy, when there may be a risk to the foetus, and late pregnancy when such fears are groundless.
And Allaah knows best.